Nonlinear Gyrokinetic Simulations Research Articles

Effects of zonal fields on energetic-particle excitations of reversed shear Alfvén eigenmode: Simulation and theory

Abstract Employing both nonlinear gyrokinetic simulation and analytical theory, we have investigated the effects of zonal (electromagnetic) fields on the energetic particle's drive of reversed shear Alfv'en eigenmodes in tokamak plasmas. Contrary to the conventional expectation, simulations with zonal fields turned on and off in the energetic particle dynamics while keeping the full nonlinear dynamics
of the thermal plasma indicate that zonal fields further enhance the instability drive and lead, thus, to a higher saturation level. These puzzling simulation results can be understood analytically in terms of the general fishbone-like dispersion relation with the correspondingly different energetic-particle phase-space structures induced by the zonal fields. Analytical expressions for the zonal fields beat driven by the reversed shear Alfv'en eigenmodes are also derived, and shown to be in good agreement with the simulation results.

Numerical dissipation induced by the low-pass filtering in nonlinear gyrokinetic simulations

De-aliasing is an essential procedure for eliminating the aliasing error in nonlinear simulations, such as nonlinear gyrokinetic turbulence simulations. An ideal approach to de-aliasing in the periodic dimension is the Fourier truncation. Finite difference low-pass filtering applied in the non-periodic direction strongly dampens aliasing modes. At the same time, it induces numerical dissipation in the region of the physically realistic solution. It is shown analytically that the long-wave dissipation coefficient is proportional to the (Np−3) power of the wavenumber under desirable constraints satisfying the highest order of accuracy, where Np is the number of filter points. Numerical results after applying the optimized low-pass filtering to the nonlinear gyrokinetic turbulence simulation suggest that the nine-point format preserves intact mesoscopic zonal structures in tokamak plasma, and is therefore suitable for long-time nonlinear turbulence simulations.

Ion-temperature- and density-gradient-driven instabilities and turbulence in Wendelstein 7-X close to the stability threshold

Electrostatic gyrokinetic instabilities and turbulence in the Wendelstein 7-X stellarator are studied. Particular attention is paid to the ion-temperature-gradient (ITG) instability and its character close to marginal stability [Floquet-type turbulence (Zocco et al., Phys. Rev. E, vol. 106, 2022, p. L013202) with no electron temperature gradient]. The flux tube version of the $\delta f$ code stella (Barnes et al., J. Comput. Phys., vol. 391, 2019, pp. 365–380) is used to run linear and nonlinear gyrokinetic simulations with kinetic electrons. The nature of the dominant instability depends on the wavelength perpendicular to the magnetic field, and the results are conveniently displayed in stability diagrams that take this dependence into account. This approach highlights the presence of universal instabilities, which are less unstable but have longer wavelengths than other modes. A quasi-linear estimate of the heat flux suggests they are relevant for transport. Close to the stability threshold, the linear eigenmodes and turbulence form highly extended structures along the computational domain if the magnetic shear is small. Numerical experiments and diagnostics are undertaken to assess the resulting radial localisation of the turbulence, which affects the interaction of the latter with zonal flows. Increasing the amplitude of the magnetic shear (e.g.through current drive) has a stabilising effect on the turbulence and, thus, reduces the nonlinear energy transport.

A new synthetic correlation electron cyclotron emission diagnostic for validating nonlinear gyrokinetic simulations of electron temperature turbulence.

To validate nonlinear gyrokinetic simulations of electron temperature turbulence, the experimental correlation electron cyclotron emission (CECE) measurements are to be compared using a synthetic CECE diagnostic, which generates modeled CECE measurement quantities by implementing realistic measurement parameters (e.g., spatial and wavenumber resolutions, radial location, etc.) to nonlinear gyrokinetic simulations. In this work, we calculate the radial and vertical spatial and wavenumber transfer functions, which are defined by the electron cyclotron emission emissivity radial profile and vertical probing antenna pattern, respectively. These transfer functions are applied to nonlinear gyrokinetic simulations of electron temperature turbulence using the continuum gyrokinetic code. A simultaneous comparison of the experimental electron temperature turbulence power spectrum and root-mean-square (RMS) level, as well as the radial correlation length with the new synthetic CECE diagnostic at a core location ρ ∼ 0.75 in an L-mode DIII-D tokamak plasma, is presented. The preliminary result shows that the synthetic CECE output underestimates the RMS level by ∼42% and overestimates the radial correlation length by ∼40%.

Nonlinear gyrokinetic modelling of high confinement negative triangularity plasmas

Nonlinear gyrokinetic simulations correctly predict particle as well as ion and electron energy fluxes of high confinement plasmas with a negative triangularity cross sectional shape, showing that core transport in these plasmas is well described by standard gyrokinetic models. Experimentally inferred power balance fluxes are mostly reproduced within one standard deviation across a wide portion of the minor radius. Experimental conditions are reproduced by ion scale simulations, without the need to include density and temperature profile curvature effects. The experimental case is used as baseline to predict that the non-dimensional confinement scaling in negative triangularity plasmas increases strongly with plasma current while slightly degrading at increasing normalized pressure and decreasing collisionality. Recent experiments showed that low toroidal rotation negatively impacts confinement; consistent with the experiment, simulations predict that low rotational shear significantly affects confinement unless the plasma effective charge is maintained above a minimum level. Core confinement is predicted to significantly degrade in low aspect ratio devices.

Enhancing predictive capabilities in fusion burning plasmas through surrogate-based optimization in core transport solvers

This work presents the PORTALS framework (Rodriguez-Fernandez et al 2022 Nucl. Fusion 62 076036), which leverages surrogate modeling and optimization techniques to enable the prediction of core plasma profiles and performance with nonlinear gyrokinetic simulations at significantly reduced cost, with no loss of accuracy. The efficiency of PORTALS is benchmarked against standard methods, and its full potential is demonstrated on a unique, simultaneous 5-channel (electron temperature, ion temperature, electron density, impurity density and angular rotation) prediction of steady-state profiles in a DIII-D ITER Similar Shape plasma with GPU-accelerated, nonlinear CGYRO (Candy et al 2016 J. Comput. Phys. 324 73–93). This paper also provides general guidelines for accurate performance predictions in burning plasmas and the impact of transport modeling in fusion pilot plants studies.

Electrostatic microturbulence in W7-X: comparison of local gyrokinetic simulations with Doppler reflectometry measurements

The first experimental campaigns of Wendelstein 7-X (W7-X) have shown that turbulence plays a decisive role in the performance of neoclassically optimized stellarators. This stresses the importance of understanding microturbulence from the theoretical and experimental points of view. To this end, this paper addresses a comprehensive characterization of the turbulent fluctuations by means of nonlinear gyrokinetic simulations performed with the code stella in two W7-X scenarios. In the first part of the paper, the amplitude of the density fluctuations is calculated and compared with measurements obtained by Doppler reflectometry (DR) in the OP1 experimental campaigns. It is found that the trend of the fluctuations along the radius is explained by the access of the DR system to different regions of the turbulence wavenumber spectrum. In the second part of the article, frequency spectra of the density fluctuations and the zonal component of the turbulent flow are numerically characterized for comparisons against future experimental analyses. Both quantities feature broad frequency spectra with dominant frequencies of O(1)–O(10) kHz.

Cross-scale interaction between microturbulence and meso-scale reversed shear Alfvén eigenmodes in DIII-D plasmas

This paper reports global nonlinear gyrokinetic simulations that couple meso-scale reversed shear Alfvén eigenmodes (RSAEs) driven by energetic particles (EPs) and ion temperature gradient (ITG) microturbulence driven by thermal plasma, using equilibrium and profiles from DIII-D discharge #159243. In simulations focusing only on the ITG, electrostatic ITG drives a huge thermal ion heat transport, which is reduced by a factor of 10 to a level close to the experimental value in electromagnetic simulation due to finite β effect. In the simulations coupling the RSAE and ITG, ITG can scatter the resonant EP nonlinearly trapped by the RSAE and damp the zonal flows generated by the RSAE. The regulation of the RSAE by the ITG greatly reduces the initial saturation amplitude of the RSAE but increases the RSAE amplitude and associated EP transport to experimental levels in the quasi-steady state. The RSAE effects on the ITG, specifically the stronger zonal flows generated by the RSAE and the RSAE frequency modulation of the ITG-induced thermal ion heat transport, in turn, leads to a reduction of the thermal ion heat transport by more than a factor of 2 . For a stronger background ITG, the regulation of the RSAE by the ITG is stronger, while the RSAE effects on the ITG are weaker. This work highlights the importance of cross-scale coupling in the dynamics of the AE turbulence and EP transport.

Investigation of fast ion effects on core turbulence in FIRE mode plasmas

Further investigation of fast ion effects on turbulence and transport in the fast ion regulated enhancement (FIRE) mode discharge (Han et al 2022 Nature 609 269–275) was performed in this work as a continuation of a previous study (Kim et al 2023 Nucl. Fusion 63 124001) that showed that the dominant turbulence suppression mechanism by fast ions is the dilution effect in the FIRE mode discharge. The current study includes (i) the impact of the fast ion relevant mode observed in the simulation of thermal energy flux, (ii) dilution effects by fast ions compared to dilution effects by other species, and (iii) fast ion effects on electron-scale turbulence. First, nonlinear gyrokinetic simulation results show that turbulence is significantly suppressed even without the fast ion relevant mode, indicating that the impact of this mode on thermal transport is not significant in this discharge. Second, further analysis on the dilution effects shows the three following results: Turbulence is not completely suppressed by the reduced main ion density fraction effect due to impurities; the reduction in energy flux can be limited by a certain impurity mode that is destabilized by a high impurity density gradient from adjusting the main ion density gradient; electrons can contribute to turbulence suppression through the main ion density gradient change, although this effect is less significant compared to other species. Third, we observe that two fast ion effects can influence the linear growth rate of the electron-scale turbulence mode. The growth rate decreases by an increase in β∗(≡(−8π/B2)dp/dr) and increases by dilution effects, suggesting that fast ion effects on electron-scale turbulence can differ depending on the operation scenario, such as the fast ion fraction. The comprehensive analysis performed in this study can enhance our understanding of fast ion physics, required for burning plasma operation in the future.

Modeling electron temperature profiles in the pedestal with simple formulas for ETG transport

This paper reports on the refinement (building on (Hatch D.R. et al 2022 Phys. Plasmas 29 062501)) and application of simple formulas for electron heat transport from electron temperature gradient (ETG) driven turbulence in the pedestal. The formulas are improved by (1) improving the parameterization for certain key parameters and (2) carefully accounting for the impact of geometry and shaping in the underlying gyrokinetic simulation database. Comparisons with nonlinear gyrokinetic simulations of ETG transport in the MAST pedestal demonstrate the model’s applicability to spherical tokamaks in addition to standard aspect ratio tokamaks. We identify bounds for model applicability: the model is accurate in the steep gradient region, where the ETG turbulence is largely slab-like, but accuracy decreases as the temperature gradient becomes weaker in the pedestal top. We use the formula to model the electron temperature profile in the pedestal for four experimental scenarios while extensively varying input parameters to represent uncertainties. In all cases, the predicted electron temperature profile exhibits extreme sensitivity to separatrix temperature and density, which has implications for core-edge integration. The model reproduces the electron temperature profile for high ηe=Lne/LTe scenarios but not for low η e scenarios in which microtearing modes (MTMs) have been identified. We develop a proof-of-concept model for MTM transport and explore the relative roles of ETG and MTM in setting the electron temperature profile.

Optimization of nonlinear turbulence in stellarators

We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC. Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.

On electromagnetic turbulence and transport in STEP

In this work, we present first-of-their-kind nonlinear local gyrokinetic (GK) simulations of electromagnetic turbulence at mid-radius in the burning plasma phase of the conceptual high-β, reactor-scale, tight-aspect-ratio tokamak Spherical Tokamak for Energy Production (STEP). A prior linear analysis in Kennedy et al (2023 Nucl. Fusion 63 126061) reveals the presence of unstable hybrid kinetic ballooning modes (KBMs), where inclusion of the compressional magnetic field fluctuation, δB∥ , is crucial, and subdominant microtearing modes (MTMs) are found at binormal scales approaching the ion-Larmor radius. Local nonlinear GK simulations on the selected surface in the central core region suggest that hybrid KBMs can drive large turbulent transport, and that there is negligible turbulent transport from subdominant MTMs when hybrid KBMs are artificially suppressed (through the omission of δB∥ ). Nonlinear simulations that include perpendicular equilibrium flow shear can saturate at lower fluxes that are more consistent with the available sources in STEP. This analysis suggests that hybrid KBMs could play an important role in setting the turbulent transport in STEP, and possible mechanisms to mitigate turbulent transport are discussed. Increasing the safety factor or the pressure gradient strongly reduces turbulent transport from hybrid KBMs in the cases considered here. Challenges of simulating electromagnetic turbulence in this high-β regime are highlighted. In particular the observation of radially extended turbulent structures in the absence of equilibrium flow shear motivates future advanced global GK simulations that include δB∥ .

Flow-shear destabilization of multiscale electron turbulence

The impact of sheared E×B flow on multiscale turbulence is studied with nonlinear gyrokinetic simulations. Simulations are based on DIII-D-like, high-confinement mode (H-mode) pedestal parameters in the regime of low ion temperature gradient drive, where there is a broad spectrum of electron temperature gradient (ETG)-driven turbulence. It is found that E×B shear can have a significant effect on ETG-driven electron transport, with an unexpected transition from a turbulence stabilization regime at moderate to large shearing rates γ E to a novel turbulence destabilization regime at low levels of γ E . In the turbulence stabilization regime, the electron energy flux decreases monotonically with γ E , even when γ E is small compared to electron mode growth rates. The stabilizing effect comes dominantly from the electron, not ion, gyrokinetic equation. In the novel destabilization regime, reduction of zonal energy results from the interaction of γ E -modulated nonlinear drive in the zonal ion gyrokinetic equation, increasing the electron transport over a broad range of shearing rates. Neither of these effects have been observed in previous electron-scale simulations.

Simultaneous reproduction of experimental profiles, fluxes, transport coefficients, and turbulence characteristics via nonlinear gyrokinetic profile predictions in a DIII-D ITER similar shape plasma

Experimental conditions obtained on the DIII-D tokamak in the ITER Similar Shape (ISS) have been compared extensively with nonlinear gyrokinetic simulation using the CGYRO code [Candy et al., J. Comput. Phys. 324, 73–93 (2016)] with comparisons spanning ion and electron heat fluxes, electron and impurity particle transport, and turbulent fluctuation levels and characteristics. Bayesian optimization techniques [Rodriguez-Fernandez et al., Nucl. Fusion 62(7), 076036 (2022)], combined with nonlinear gyrokinetics, have been used to obtain simultaneously Qi, Qe, and Γe flux-matched profiles that are found to be in good agreement with experimental profile measurements. Synthetic diagnostics were used to compare measured beam emission spectroscopy and correlation electron cyclotron emission turbulent fluctuations with nonlinear simulation. Although some disagreements exist, nonlinear simulations are found to be in generally good agreement with measured fluctuation levels, spectral shapes, and measured radial trends in low-k δne/ne and δTe/Te. Low (Li and C) and mid-Z (Ca) impurity transport was also compared with these flux-matched simulations. Fully stripped, low-Z impurities are well reproduced by the gyrokinetic modeling while clear disagreement exists in comparisons with mid-Z impurities. Nonlinear gyrokinetic investigation into the Z dependence of impurity transport in the ISS conditions is also performed, demonstrating clear trends of impurity diffusion with impurity Z (both D∝Z and D∝1/Z) that vary with the radial location studied. These trends are shown to result from the local dominance of ion temperature gradient or ∇n driven trapped electron mode turbulence and may contribute to the disagreement between simulation and experiment in mid-Z impurity transport. The results of this work represent one of the most complete validation studies of the gyrokinetic model performed to date and provide an example of new capabilities for predicting performance in future fusion devices.

Global linear and nonlinear gyrokinetic simulations of tearing modes

To better understand multi-scale interactions between global tearing modes and microturbulence in the Madison Symmetric Torus (MST) reversed-field pinch (RFP), the global gyrokinetic code Gene is modified to describe global tearing mode instability via a shifted Maxwellian (SM) distribution consistent with experimental equilibria. The implementation of the SM is tested and benchmarked by comparisons with different codes and models. Good agreement is obtained in code-code and code-theory comparisons. Linear stability of tearing modes of a non-reversed MST discharge is studied. A collisionality scan is performed to the lowest order unstable modes (n = 5, n = 6) and shown to behave consistently with theoretical scaling. The nonlinear evolution is simulated, and saturation is found to arise from mode coupling and transfer of energy from the most unstable tearing mode to small-scale stable modes mediated by the m = 2 tearing mode. The work described herein lays the foundation for nonlinear simulation and analysis of the interaction of tearing modes and gyroradius-scale instabilities in RFP plasmas.

Self-Organized Evolution of the Internal Transport Barrier in Ion-Temperature-Gradient Driven Gyrokinetic Turbulence.

Understanding the self-organization of the most promising internal transport barrier in fusion plasmas needs a long-time nonlinear gyrokinetic global simulation. The neighboring equilibrium update method is proposed, which solves the secularity problem in a perturbative simulation and speeds up the numerical computation by more than 10 times. It is found that the internal transport barrier emerges at the magnetic axis due to inward propagated turbulence avalanche, and its outward expansion is the catastrophe of self-organized structure induced by outward propagated avalanche.

Complex structure of turbulence across the ASDEX Upgrade pedestal

The theoretical investigation of relevant turbulent transport mechanisms in H-mode pedestals is a great scientific and numerical challenge. In this study, we address this challenge by global, nonlinear gyrokinetic simulations of a full pedestal up to the separatrix, supported by a detailed characterisation of gyrokinetic instabilities from just inside the pedestal top to the pedestal centre and foot. We present ASDEX Upgrade pedestal simulations using an upgraded version of the gyrokinetic, Eulerian, delta-f code GENE (genecode.org) that enables stable global simulations at experimental plasma $\beta$ values. The turbulent transport is found to exhibit a multi-channel, multi-scale character throughout the pedestal with the dominant contribution transitioning from ion-scale trapped electron modes/micro-tearing modes at the pedestal top to electron-scale electron temperature gradient modes in the steep gradient region. Consequently, the turbulent electron heat flux changes from ion to electron scales and the ion heat flux reduces to almost neoclassic values in the pedestal centre. $E\times B$ shear is found to strongly reduce heat flux levels in all channels (electron, ion, electrostatic, electromagnetic) and the interplay of magnetic shear and pressure gradient is found to locally stabilise ion-scale instabilities.

Absolute value measurement of ion-scale turbulence by two-dimensional phase contrast imaging in Large Helical Device

Absolute value measurements of turbulence amplitude in magnetically confined high-temperature plasmas can effectively explain turbulence-driven transport characteristics and their role in plasma confinements. Two-dimensional phase contrast imaging (2D-PCI) is a technique to evaluate the space-time spectrum of ion-scale electron density fluctuation. However, absolute value measurement of turbulence amplitude has not been conducted owing to the nonlinearity of the detector. In this study, the absolute measurement method proposed in the previous study is applied to turbulence measurement results in the large helical device. As a result, the localized turbulence amplitude at ne = 1.5 × 1019 m-3 is approximately 3.5 × 1015 m-3, which is 0.02% of the electron density. In addition, the evaluated poloidal wavenumber spectrum is almost consistent, within a certain error range, the spectrum being calculated using a nonlinear gyrokinetic simulation. This result is the first to the best of our knowledge to quantitatively evaluate turbulence amplitudes measured by 2D-PCI and compare with simulations.

A self-sustaining mechanism for internal transport barrier formation in HL-2A tokamak plasmas

The formation of Internal Transport Barrier (ITB) is studied in HL-2A plasmas by means of nonlinear gyrokinetic simulations. A new paradigm for the ITB formation is proposed in which different physics mechanisms play a different role depending on the ITB formation stage. In the early stage, fast ions, introduced by Neutral Beam Injection ion system, are found to stabilize the thermal-ion-driven instability by dilution, thus reducing the ion heat fluxes and finally triggering the ITB. Such dilution effects, however, play a minor role after the ITB is triggered as electromagnetic (EM) effects are dominant in the presence of established high pressure gradients. We define the concept of ITB self-sustainment, as the low turbulence levels found within the fully formed ITB are consequences of large scale zonal flows, which in turn are fed by a non-linear interplay with large scale high frequency EM perturbations destabilized by the ITB itself.

Turbulence stabilization in tokamak plasmas with high population of fast ions

This letter provides a new physical insight into the fast ion effects on turbulence in plasmas having a high fast ion fraction and peaked fast ion density profile. We elucidate turbulence stabilization mechanisms by fast ions that result in internal transport barrier formation in the fast ion regulated enhancement mode plasma. Both linear and nonlinear gyrokinetic simulations show that the dominant turbulence suppression mechanisms are the dilution effects. In particular, we find that turbulence can be sufficiently suppressed solely by an inverted main ion density gradient due to fast ions, for the first time. New physical findings reported here improve our understanding of fast ion effects on turbulence, essential for fusion energy production where . Moreover, they will open up a new methodology to control plasma turbulence applicable to a wide range of plasma confinement regimes.